An ecosystem with mature trees has a relatively constant population of beetles $($species $1)$ that number around $109.$ At t$=0,$ an invasive beetle species $($species $2)$ is introduced accidentally. If this population initially grows at the exponential rateN$2($t$)=$ert,where$$r$=0\mathrm{.}2$per monthhow long will it take for species $2$ to overtake the population of the resident species $1?$